Question
Let $$f\left( x \right) = x,\,g\left( x \right) = \frac{1}{x}$$ and $$h\left( x \right) = f\left( x \right)g\left( x \right).$$ Then, $$h\left( x \right) = 1$$ if and only if :
A.
$$x$$ is a real number
B.
$$x$$ is a rational number
C.
$$x$$ is an irrational number
D.
$$x$$ is a non-zero real number
Answer :
$$x$$ is a non-zero real number
Solution :
$$\eqalign{
& D\left( f \right) = R,\,D\left( g \right) = R - \left\{ 0 \right\} \cr
& \therefore \,D\left( h \right) = R - \left\{ 0 \right\}{\text{ and }}h\left( x \right) = f\left( x \right)g\left( x \right) = x \times \frac{1}{x} = 1 \cr
& \therefore \,h\left( x \right) = 1{\text{ if and only if }}x\, \in \,R - \left\{ 0 \right\} \cr} $$